MathDB

1

Part of 1992 National High School Mathematics League

Problems(2)

Parabola Problem

Source: 1992 National High School Mathematics League, Exam One, Problem 1

2/26/2020
For any positive integer nn, AnA_n and BnB_n are intersection of parabola y=(n2+n)x2−(2n+1)x+1y=(n^2+n)x^2-(2n+1)x+1 and xx-axis. Then, the value of ∣A1B1∣+∣A2B2∣+⋯+∣A1992B1992∣|A_1B_1|+|A_2B_2|+\cdots+|A_{1992}B_{1992}| is (A)19911992(B)19921993(C)19911993(D)19931992\text{(A)}\frac{1991}{1992}\qquad\text{(B)}\frac{1992}{1993}\qquad\text{(C)}\frac{1991}{1993}\qquad\text{(D)}\frac{1993}{1992}
conicsparabola
Geometry~

Source: 1992 National High School Mathematics League, Exam Two, Problem 1

2/27/2020
A1A2A3A4A_1A_2A_3A_4 is cyclic quadrilateral of ⊙O\odot O. H1,H2,H3,H4H_1,H_2,H_3,H_4 are orthocentres of △A2A3A4,△A3A4A1,△A4A1A2,△A1A2A3\triangle A_2A_3A_4,\triangle A_3A_4A_1,\triangle A_4A_1A_2,\triangle A_1A_2A_3. Prove that H1,H2,H3,H4H_1,H_2,H_3,H_4 are concyclic, and determine its center.
geometrycyclic quadrilateral